Descriptive Statistics III
REVIEW
• Variability
• Range, variance, standard deviation
S2 
 X
 M
n  1
2
S 
S
2
• Coefficient of variation (S/M): 2 data sets
• Value of standard scores?
X M
Z 
S
T  50  10 z
p  50  z ( percentile )
Correlation and Prediction
HPHE 3150
Dr. Ayers
Variables
Dependent
•
•
•
•
•
•
(ordinal/continuous: #)
Presumed effect
Consequence
Measured by researcher
Predicted
Criterion
Y
Y
dv
X
iv
Independent
•
•
•
•
•
•
(categorical: name)
Presumed cause
Antecedent
Manipulated by researcher
Predicted from
Predictor
X
Correlation
(Pearson Product Moment or r)
•Are two variables related?
•Car speed & likelihood of getting a ticket
•Skinfolds & percent body fat
•What happens to one variable when the other one
changes?
•Linear relationship between two variables
•1 measure of 2 separate variables or 2 measures of 1 variable
•Provides support for a test’s validity and reliability
Attributes of r
magnitude & direction
N
e
g
a
t
i
v
e
P
o
s
i
t
i
v
e
1
.
0
0
.
7
0
0
.
3
0
0
0
.
3
0
0
.
7
0
1
.
0
Perfect High
Low
Zero Low
High Perfect
Scatterplot of correlation between
pull-ups and chin-ups
Chin-ups (#completed)
(direct relationship/+)
16
14
12
10
8
6
4
2
0
0
2
4
6
8
Pull-ups (#completed)
10
12
14
Scatterplot of correlation between
body weight and pull-ups
Pull-ups (#completed)
(indirect/inverse relationship/-)
14
12
10
8
6
4
2
0
120
130
140
150
Weight (lb)
160
170
180
Scatterplot of zero correlation (r = 0)
Figure 4.4
Y
8
6
4
2
0
0
2
4
6
X
8
10
Correlation Formula
(page 60)
r

n  X

n  XY –  X Y 
2
–  X    n Y – Y  
2
2
2
Correlation issues
• Correlation ≠ causation
• -1.00 < r < +1.00
• Coefficient of Determination (r2) (shared variance)
• r=.70
r2=.49
49% variance in Y accounted for by X
Y
dv
X
iv
• Negative correlation possibly due to:
• Opposite scoring scales
• True negative relationship
• Linear or Curvilinear (≠ no relationship; fig 4.6)
• Range Restriction (fig 4.7; ↓ r)
• Prediction (relationship allows prediction to some degree)
• Error of Prediction (for r ≠ 1.0)
• Standard Error of Estimate (prediction error)
Limitations of r
Figure 4.6
Curvilinear relationship
Example of variable?
Figure 4.7
Range restriction
Limitations of r
Correlation & Prediction I
REVIEW
• Bivariate nature of correlations
• X (iv) & Y (dv)
• +/- relationships
• Range of r?
• Coefficient of Determination (r2) (shared variance)
• Coefficient of variation (S/M): 2 data sets
• Low V (.1-.2=homo): M accounts for most variability in scores
• Curvilinear relationship?
• Correlation/Causation?
Fitness/PA
Uses of Correlation
• Quantify RELIABILITY of a test/measure
• Quantify VALIDITY of a test/measure
• Understand nature/magnitude of bivariate relationship
• Provide evidence to suggest possible causality
Misuses of Correlation
• Implying cause/effect relationship
• Over-emphasize strength of relationship due to
“significant” r
Correlation and prediction
% Fat
Skinfolds
Sample Correlations
Excel document
Standard Error of Estimate
(SEE)
Average error in the process of predicting Y from X
Standard Deviation of error
S e  Sy 1  r
As r ↑, error ↓
As r ↓, error ↑
Is ↑r good? Why/Not?
Is ↑ error good? Why/Not?
2